A summing amplifier can be used as a digital-to-analog converter (DAC). An example of a 4-bit DAC is shown in Figure P9.37. When switch \(S_{3}\) is connected to the \(-5 \mathrm{~V}\) supply, the most significant bit is \(a_{3}=1\); when \(S_{3}\) is connected to ground, the most significant bit is \(a_{3}=0\). The same condition applies to the other switches \(S_{2}, S_{1}\), and \(S_{o}\), corresponding to bits \(a_{2}, a_{1}\), and \(a_{o}\), where \(a_{o}\) is the least significant bit.

(a) Show that the output voltage is given by

\[v_{O}=\frac{R_{F}}{10}\left[\frac{a_{3}}{2}+\frac{a_{2}}{4}+\frac{a_{1}}{8}+\frac{a_{o}}{16}\right]\]

where \(R_{F}\) is in \(\mathrm{k} \Omega\).

(b) Find the value of \(R_{F}\) such that \(v_{O}=2.5 \mathrm{~V}\) when the digital input is \(a_{3} a_{2} a_{1} a_{o}=1000\).

(c) Using the results of part (b), find \(v_{o}\) for: (i) \(a_{3} a_{2} a_{1} a_{o}=0001\), and (ii) \(a_{3} a_{2} a_{1} a_{o}=1111\).

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9 VR=-5 V RF ww R3 = 20 k2 ww R = 40 k www R = 80 k ww R = 160 k == www Figure P9.37